SMALL AMPLITUDE NONLINEAR DUST ACOUSTIC WAVE PROPAGATION IN SATURN S F, G AND E RINGS Letter to the Etor SAMIRAN GHOSH, TUSHAR K. CHAUDHURI, S. SARKAR, MANORANJAN KHAN an M.R. GUPTA Centre for Plasma Stues, Faculty of Scence, Jaavpur Unversty, Calcutta 700 032, Ina (Receve 8 September 2000; accepte 3 March 2001) Abstract. Nonlnear propertes of small ampltue ust acoustc waves, ncorporatng both the on nertal effect an ust rft effect have been stue. The effect of ust charge varaton s also ncorporate. It s seen that ue to the ust charge varaton, a Korteweg-e Vres (KV) equaton wth postve or negatve ampng term epenng on the wave velocty an the rng parameters governes the nonlnear ust acoustc wave. It s seen that the ampng or growth arses ue to the assumpton that ust hyroynamcal tme scale s much smaller than that of the ust chargng scale. Ths assumpton s val only for planetary rngs such as Saturn s F, G an E rngs. Numercal nvestgatons reveal that all the three rngs n F, G an E, ust acoustc soltary wave amts both negatve an postve potentals. Instablty arses from the avalable free energy of rft moton of ust grans only for the wave wth wave velocty λ<v 0, the rft velocty of the ust. 1. Introucton Dust grans now appear to be almost ubqutous n astrophyscal plasma envronments. Dust partcles n plasma wll normally be electrcally charge ue to plasma currents, photo emsson, seconary emsson an fel emsson (Drane an Salpeter, 1979; Whpple, 1981; Horany, 1996). In plasma, charge varatons of the ust partcles may lea to resonance, ffuson an rft phenomena (Burns an Schaffer, 1989). In the lnear theory, several authors (Rao et al., 1990; e Angels et al., 1988; D Angelo, 1990; Shukla an Bharuthram, 1991; Rosenberg, 1993; Mens an Rosenberg, 1994) have stue the low frequency characterstcs of usty plasmas havng fxe charge ust grans. But n the lnear theory, ue to the conseraton of charge varaton of the ust partcles, the wave become ampe (Shukla an Stenflo, 1992; Varma et al., 1993; Melanso et al., 1993; Melanso et al., 1993). Among these works, Shukla et al. (1991), Rosenberg (1993) an Melanso et al. (1993a, b) have consere ust rft velocty between the ust an the plasma to stuy the lnear characterstcs of ust acoustc wave. In the nonlnear theory, several authors (Bharuthram an Shukla, 1992; Verheest, 1992; Ynhua an Yu, 1994; Ghosh et al., 2000) have stue the nonlnear E-mal: sran@jufs.ernet.n or sran g@yahoo.com Dept. Of Apple Mathematcs, Unversty Of Calcutta 700 009 E-mal: mk@jufs.ernet.n Astrophyscs an Space Scence 278: 463 475, 2001. 2001 Kluwer Acaemc Publshers. Prnte n the Netherlans.
464 S. GHOSH ET AL. ust acoustc waves for the fxe charge ust grans. Ther analyses have shown that nonlnear ust acoustc wave can form soltons of ether postve or negatve electrostatc potentals. Ma an Lu (1997) an Xe et al. (1998) have stue the nonlnear ust acoustc wave wthout ust rft, conserng gran charge varaton uner the assumpton that the ust hyroynamcal tme scale s much greater than that of the ust chargng tme scale, whch s observe n experments (Barkan et al., 1995; Wnske an Jones, 1995) by sageev potental metho. However, n astrophyscal plasmas lke planetary rngs the stuaton s qute opposte,.e. n planetary rngs the ust hyroynamcal tme scale s much smaller than that of the ust chargng tme scale. Also, n the nonlnear theory, the charge varaton of the ust partcles may lea to the formaton of shock wave (Popel et al., 1996; Popel et al., 1998) n the usty plasma. In ths paper, the nonlnear propertes of small ampltue ust acoustc wave nclung ust charge varaton n Saturn s F, G an E rngs have been stue by the reuctve perturbaton technque. The ust rft velocty V 0 an the on thermal velocty V th are also consere, because n Saturn s F, G an E rngs, V 0 s comparable to (Whpple, 1981; Northrop et al., 1989; Melanso et al., 1993) V th an the ncluson of V 0 an V th mofes the expresson for on current. It s seen that n all the three rngs F, G an E nonlnear ust acoustc soltary wave amts both postve an negatve potentals. It s also seen that n Saturn s F, G an E rngs, a KV equaton wth postve or negatve ampng term epenng on the physcal characterstcs of the rng an also the wave velocty governs the nonlnear ust acoustc soltary ( wave. Ths ) happens because n these rngs the ust hyroynamcal tme scale τ ω 1 p s much smaller than that of the ust chargng tme scale ( τ ν 1 ). Secton 2 contans the moel an the basc equatons escrbng the moel. Secton 3 contans the reuctve perturbaton analyss an the nonlnear KV equaton. Numercal results an scussons are gven n Secton 4 an n Secton 5 respectvely. 2. Basc Equatons We conser a three component collsonless, non relatvstc, unmagnetze usty plasma cocsstng of hghly charge ust grans, ons an electrons. The charge varaton of the ust partcles, the ust rfts an on nerta, are also consere. In ths stuaton the charge neutralty conton s n 0 = n e0 + z n 0 (1) where n 0,n 0 an n e0 are the equlbrum number enstes of ons, ust grans an electrons. We assume that the wave vector les along the X recton an the ust rft velocty s also along the X recton. In ths stuaton the basc equatons are the
SMALL AMPLITUDE NONLINEAR DUST ACOUSTIC WAVE PROPAGATION 465 one mensonal contnuty an momentum flu equatons for the ust grans an ons an Posson s equaton. In normalze form these equatons can be wrtten as N T + (N V ) = 0 (2) X V T + V V X = ( Q 1) X N T + (N V ) X = 0 (4) ) µ ( V T + V V X = 1 σ X 1 N N X 2 X 2 = λ2 e2 ɛ 0 T e [n 0 N n e + z n 0 ( Q 1) N ] (6) n e = n e0 exp ( ) (7) [ ] ( Q 1) Q = Q z s the normalze ust charge, normalze by z e e.the ust charge s gven by Q = z e + Q. The term µ = z m σm (3) (5) represents the on nerta. N j = n j n j0 (j =, ), n j0 s the equlbrum number enstes of the jth speces. V j (j =, ) s the partcle velocty of the jth speces normalze by ust acoustc spee c = z kt e m an s the electrostatc wave potental normalze by kt e e. σ = T T e, T an T e are the on an electron temperatures respectvely. The tme an space scales are ω 1 p an λ so that the spatal coornate X an tme T are normalze by ust Debye length λ = an ust plasma frequency z n 0 e 2 ɛ 0 m ɛ0 kt e z n 0 e 2 ω p = respectvely. In ths paper the on nerta (m = 0) s nclue because ust rft an on nerta plays an mportant role for the propagaton of ust acoustc wave n planetary rng. To obtan the normalze ust charge varaton Q, we conser the orbtal moton lmte current balance equaton, n normalze form, t reas as ( ) z e Q T + V Q = τ (I e + I ) (8) X The ust hyroynamcal tme scale τ an the ust chargng scales (Vlamrov, 1994) are gven as follows [ ] 1 (9) τ ω 1 p ; a 2π ω 2 p V th (χ (y) + (z + 1) ψ (y))
466 S. GHOSH ET AL. ω p s the on plasma frequency. The electron an on current (Whpple, 1981; Northrop et al., 1989) are gven by I e = πa 2 8T e e n e0 exp ( + z ( Q 1)) (10) πm e I = πa 2 8T ( e n e0 N χ (y) z ) πm σ ψ (y)( Q 1) (11) where χ (y) = 1 4 π ( 1 + 2y 2 ) y 1 erf (y) + 1 2 exp ( y 2) (12) ψ (y) = 1 2 πy 1 erf (y) ; erf (y) = 2 π y 0 exp ( x 2) x (13) where y = V 0 V th, V 0 an V th nare the ust rft velocty an on thermal veloctes, respectvely. z = z e 2 4πɛ 0 at e,4πɛ 0 a s the capactance of the sphercal ust gran of raus a. 3. Reuctve Perturbaton Analyss In orer to stuy the nonlnear propagaton of small ampltue ust acoustc wave, usng the stanar reuctve perturbaton technque, the nepenent varables can be stretche as ξ = ɛ (X λt ) ; τ = ɛ 3 2 T (14) where λ s the phase velocty of the lnear ust acoustc waves an ɛ s a small parameter characterzng the strength of the nonlnearty. The epenent varables are expane as N = 1 + ɛn (1) + ɛ 2 N (2) + (15) V = V 0 + ɛv (1) + ɛ 2 V (2) + (16) N = 1 + ɛn (1) + ɛ 2 N (2) + (17) V = ɛv (1) + ɛ 2 V (2) + (18) = ɛ (1) + ɛ 2 (2) + (19)
SMALL AMPLITUDE NONLINEAR DUST ACOUSTIC WAVE PROPAGATION 467 Q = ɛ Q (1) + ɛ 2 Q (2) + (20) To make the nonlnear perturbaton consstent we assume that the rato τ s proportonal to ɛ 3 2, uner the assumpton that the ust hyroynamcal tme scale s much smaller than that of ust chargng tme scale, where τ 1 1 (χ (y) σ + ψ (y)(z + 1)) (21) µ (1 δ)(1 + δσ) 2π Thus τ νɛ 3 2 (22) ν s a fnte quantty of the orer of unty. Later we shall show that ths assumpton s well justfe for Saturn s rng. The bounary constons are as follows, as X N,N 1; V V 0 ;, V, Q 0 (23) Now ntroucng (14) (20) an (22) n to (2) (8) an equatng coeffcents of terms n lowest orer n ɛ, we obtan the followng relatons V (1) = (λ V 0 ) N (1) (24) V (1) = (1) (λ V 0 ) (25) V (1) = λn (1) (26) V (1) = ( ) (1) + N (1) σ λµ (27) (1 δ) N (1) = N (1) δ (1) + (1 δ) Q (1) (28) δ = n e0 n 0 (29) (λ V 0 ) Q (1) ξ = 0 (30) Now by vrtue of the bounary contons that perturbatons vansh at X = (ξ = ) for all tme slow or fast, equaton (30) mplyng Q (1) = Q (1) (τ) = 0 (31)
468 S. GHOSH ET AL. The vanshng of the ust charge fluctuaton Q (1) to orer O (ɛ) s thus a consequence( of) the assumpton that the ust hyroynamcal tme τ s smaller by orer O ɛ 2 3 than that of ust chargng tme. From (24) an (25) we obtan N (1) = (1) (λ V 0 ) 2 (32) From (26) an (27) we obtan N (1) = (1) σ ( λ 2 µ 1 ) (33) Fnally from (28), usng (31), (32) an (33) we obtan the followng bquaratc equaton n λ as 1 σ ( (1 δ) ) + λ 2 µ 1 (λ V 0 ) 2 = δ (34) where a 0 λ 4 + 4a 1 λ 3 + 6a 2 λ 2 + 4a 3 λ + a 4 = 0 (35) a 0 = µ δ (1 δ),a 1 = µ δv 0 2 (1 δ),a 2 = 1 [ δv0 2 6 (1 δ) µ ] (1 + δσ), (1 δ) (1 + δσ) a 3 = 2 (1 δ),a (1 + δσ) 4 = 1 (1 δ) V 0 2 (36) For δ = n e0 n 0 0.e. for an effectvely two components plasma comprsng of postve ons an negatvely charge ust grans wth fxe charge an T e T,the roots of equaton (34) are as follows V 0 ± V0 2 λ = (1 + µ ) ( V0 2 1) (37) 1 + µ But µ 1 an hence the roots of (37) becomes λ V 0 ± 1 µ V0 2 (38) Equaton (38) shows that λ s real only f 1 V 0 = V th (39) µ
SMALL AMPLITUDE NONLINEAR DUST ACOUSTIC WAVE PROPAGATION 469 Therefore for the two component ust on plasma, the ust acoustc soltary wave exsts only f the ust rft velocty s less than the on thermal velocty (Ghosh et al., 2000). Now equatng the coeffcents of terms n next hgher orer n ɛ we obtan the followng relatons N (1) τ + N (1) V (1) ξ + N (1) ξ V (1) = (λ V 0 ) N (2) ξ V (2) ξ (40) V (1) τ + V (1) V (1) ξ + Q (1) (1) ξ = (λ V 0 ) V (2) ξ + (2) ξ (41) N (1) τ + N (1) V (1) ξ + N (1) ξ V (1) = λn (2) ξ V (2) ξ (42) ( ) µ V (1) τ + V (1) V (1) ξ λn (1) V (1) ξ + 1 σ N (1) (1) ξ = λµ V (2) ξ 1 σ (2) ξ N (2) ξ (43) ( δ ξξ = 1 δ ( 1 1 δ (1) ) (2 ) + 1 ( ) δ (1)2 2 1 δ ) N (2) + N (2) Q (2) N (1) Q(1) (44) ( Q (2) β σλ 2 µ 2 ) = ν σ ( λ 2 µ 1 ) (λ V 0 ) 2 (1) (45) χ (y) σ + ψ (y) z β = (46) z (χ (y) σ + (z + 1) ψ (y)) The expressons (45) s obtane usng equaton (31) an (33). The subscrpt ξ an τ enote fferentaton wth respect to ξ an τ respectvely. Usng (24), (32) an (31), from equatons (40) an (41) we obtan (1) ξ N (2) ξ = 2 (1) τ (λ V 0 ) 3 + 3 (1) (λ V 0 ) 4 (λ V 0 ) 2 (47) Smlarly usng (26), (33) from equatons (42) an (43) we obtan N (2) ξ = 2λµ ( (1) τ 3λ 2 σ ( λ 2 µ 1 ) 2 µ 1 ) (1) (1) ξ σ ( 2 λ 2 µ 1 ) (2) ξ + 3 σ ( λ 2 µ 1 ) (48) Dfferentatng equaton (44) wth respect to ξ an nsertng equaton (31), (34), (47) an (48) we obtan the followng mofe KV equaton as (1) τ α (1) (1) ξ + β (1) ξξξ + γ (1) = 0 (49) (2) ξ
470 S. GHOSH ET AL. where [ α = β 3 (λ V 0 ) 4 + ( ) δ 1 δ ( 3λ 2 µ 1 ) σ 2 (1 δ) ( λ 2 µ 1 ) 3 ] (50) [ ] β = 1 1 2 2 (λ V 0 ) 3 + λµ σ (1 δ) ( λ 2 µ 1 ) (51) 2 ( ββ σλ 2 µ 2 ) γ = ν σ ( λ 2 µ 1 ) (λ V 0 ) 2 (52) Employng Karpman an Maslov s metho an usng bounary contons (23) we arrve at the followng approxmate soluton whch exhbts the slow tme (.e. τ epenent) evoluton of the ampltue, the wth an the velocty of the sech 2 soltary wave (1) = (1) (τ) sech 2 α (1) (τ) η (53) 12β where η = ξ Uτ = ɛ [X (λ + ɛu)] (54) (1) (τ) = (1) (0) e γτ ; U = α 6 (1) (0) e γτ (55) (1) (τ) an U (τ) are the τ epenent solton ampltue an solton velocty. Several features of physcal nterest arse epenng on the values of the coeffcents α, β an γ of the KV equaton (49). These are functons of λ, the roots of the sperson equaton (34) an also other plasma parameters. From (53) t s event that we must have α β (1) (τ) > 0 (56) The last equaton an equaton (32) have followng consequences α β > 0 (1) < 0 N (1) > 0 α β < 0 (1) > 0 N (1) < 0 (57) Another feature of nterest s that the wave ampltue (1) (τ) may grow or may ecay epenng on the algebrac sgn of γ.as 1 µ < 2 σµ (σ = T T e 1 n practcally
SMALL AMPLITUDE NONLINEAR DUST ACOUSTIC WAVE PROPAGATION 471 TABLE I Dfferent Parameters In Saturn s F, G an E rngs. (Column 1 an 2: Goertz, 1998; Goertz an Shan, 1989, Column 3: Melanso et al., 1993, Column 5: Havnes et al., 1990) ( Rngs δ n m 3) ( V rft ms 1) V 0 ( 10 3) ( P 10 3) y = V ( 0 V µ th 10 7 ) ( τ 10 4) F 0.01 3.2 10 7 6.4 10 3 2.324 2.2 0.18 0.06427 3.3 G 0.99 3 10 1 1.3 10 4 0.7769 2.1 0.37 2.3204 9.6 E 0.99 3 10 1 2 10 4 1.2105 2.1 0.58 2.338 9.3 all cases an n partcular σ = 1 for Saturn s rngs) the conton for growth γ<0 s satsfe n ether of the followng three cases β<0, λ 2 < 1 ( < 2 ) (58) µ σµ β<0, λ 2 > 2 σµ > 1 µ (59) 1 β>0, <λ 2 < 2 (60) µ σµ In the next secton t wll be seen that t s only the conton (58) whch apples for all the rngs of Saturn. The root of the equaton (34) oes not exst, satsfyng ether (59) or (60). Numercal calculatons an ther physcal consequences are presente n the next secton. 4. Numercal Results We assume that there exst a corotatng plasma wth O + ons n Saturn s rngs an the electron on temperature rato σ = 1. To apply the results of the prevous secton to the fferent rngs of Saturn, we nee the values of the varous plasma parameters as lste n Table I. For all the three rngs of Saturn σ = T T e = 1, a = 1µm, n = 10 7 m 3 an T e = 10 2 ev (Goertz an Shan, 1988; Goertz, 1989). Column 5 of Table I gves the value of the mensonless parameter (Havnes et al., 1990) P = 6.95 10 8 a µ T ev n 0 n e0
472 S. GHOSH ET AL. TABLE II Coeffcents of nonlnearty, sperson an ampng or growth rate n fferent rngs for fferent wave veloctes λ Rngs Roots α β γ F λ 1 = 1.254 10 5 6.019 10 4 6.144 10 4 6.938 10 3 λ 2 = 2.325 10 3 4.976 10 1 9.775 10 1 13.709 λ 3 = 2.322 10 3 4.8501 10 1 9.667 10 13.658 G λ 1 = 2.943 10 3 2.906 10 4 7.4704 3.518 10 8 λ 2 = 7.775 10 2 1.671 5.704 10 2 7.249 10 1 λ 3 = 7.765 10 2 2.047 1.465 10 2 4.618 10 1 E λ 1 = 2.932 10 3 2.895 10 3 7.441 5.519 10 8 λ 2 = 1.2107 10 3 3.262 5.718 10 3 3.927 10 1 λ 3 = 1.2102 10 3 3.746 2.523 10 3 2.991 10 1 where a µ s the ust raus n mcron an T ev electron temperature n electron volts. The value of the rato of the ust hyroynamcal tme scale τ to the ust chargng tme scale has been calculate usng a λ 10 6 (e Angels, 1992) an the approprate parameter values gven n Table I. It s foun that τ 3.3 10 4, 9.6 10 4 an 9.3 10 4 for the F, G an E rngs respectvely. These values justfy the scalng τ νɛ 3 2, on the bass of whch chargng equaton s approxmate an the subsequent analyss has followe. Table I also shows that for all the three rngs F, G an E V 0 < 1 (61) Uner the above conton all the four roots of the sperson equaton (34) are real. Three of them are postve whle the fourth one s negatve. For all the rngs they occur n orer gven below λ 4 < 0 <λ 3 <V 0 <λ 2 < 1 <λ 1 (62) The roots λ 1,λ 2,λ 3 an λ 4 of Equaton (34) are calculate usng values of δ, V 0 an µ as gven n Table I. The KV equaton (49) coeffcents α, β an γ gven by Equatons (50), (51) an (52) are then obtane corresponng to λ 1,λ 2 an λ 3. These are shown n Table II. From the expresson for β gven by (51) t s seen that β<0spossble only f λ<v 0.AsV 0 < 1 (Equaton (61)) ths mmeately rules out the possblty of (59) beng true. Moreover from (62) t s clear that Saturn s rngs there oes not
SMALL AMPLITUDE NONLINEAR DUST ACOUSTIC WAVE PROPAGATION 473 1 exst any root n the nterval µ <λ 2 < 2 σµ an consequently (60) can not apply. Thus only the growth conton (58) may hol goo an t actually oes for the wave wth wave velocty λ = λ 3 <V 0 < 1. Thus n F rng α β > 0forλ = λ 1 > 1 an corresponng to (57) the soltary wave has a negatve potental an so t s a ust ensty conensaton an (33) on ensty epleton wave. Moreover γ has a very small postve value for λ = λ 1 whch mples that the wave s allmost unampe. On the other han for λ = λ 2 < 1,wefnthat α < 0forFrnganγ s seen to have a large postve value. β Hence t s a strongly ampe postve potental wave wth ust ensty epleton an as ( λ 2 µ 1 ) < 0, the on ensty s also eplete (33). The G rng scenaro s qualttavely the same as the E rng one. For λ = λ 1 > 1, t s a very weakly, allmost unampe postve potental wave wth N (1) < 0 an N (1) > 0as α β < 0anγ>0 wth extremely small magntue. For λ = λ 2 < 1 the algebrac sgns of α β an γ are same wth γ 10 1. So t s a moerately ampe negatve potental wave wth both ust an on ensty eplete. For λ = λ 3 (0 <λ 3 <V 0 ),wefnthatγ has a large negatve value n F rng, whereas t s of moerate magntue n G an E rngs. Thus for F rng waves wth λ = λ 3 s strongly growng, whereas for G an E rngs waves wth λ = λ 3 are moerately growng. The nstablty arses from the avalable free energy of rft moton of ust grans, the rft velocty V 0 beng greater than the wave velocty λ 3. The opposte phenomenon,.e., ampng occurs for waves wth wave velocty λ = λ 1,λ 2 >V 0. 5. Dscusson In ths paper, the nonlnear ust acoustc wave propagaton characterstcs n the planetary rng such as Saturn s F, G an E rngs have been stue. (a) It s seen that n all the three rngs of Saturn τ 1.e. the ust chargng frequency s much smaller than the ust plasma frequency. (b) The ust rft velocty an on nertal effects are ncorporate n ths nvestgaton. (c) The effect of ust charge varatons arsng ue to small but fnte value of the rato τ s consere. () The lnear sperson equaton s a bquaratc n the ust wave velocty. For ust rft velocty V 0 < 1, the on thermal velocty a conton whch always apples to Saturn s rngs, all the four roots are real an they occur n the orer ncate n Equaton (62). (e) Numercal nvestgatons reveal that ust acoustc soltary wave amts both postve an negatve potental wth corresponng ust an on ensty enhancement or epleton accorng to (32), (33) n all the three rngs an these may ecay very weakly or may be nonlnearly unstable wth moerate to strong growth
474 S. GHOSH ET AL. TABLE III Characterstcs of the nonlnear ust acoustc waves wth fferent wave veloctes λ n F, G an R rngs Dfferent Rngs F G E λ 1 > 1 (1) < 0 (1) > 0 (1) > 0 allmost unampe allmost unampe allmost unampe V 0 <λ 2 < 1 (1) > 0 (1) < 0 (1) < 0 strong ampe moerate ampe moerate ampe 0 <λ 3 <V 0 (1) < 0 (1) > 0 (1) > 0 strong growth moerate growth moerate growth rate epenng on whether the wave velocty s greater or less than the ust rft velocty. (f) The physcal characterstcs of nonlnear ust acoustc waves are smlar n G an E rngs, whle for the F rng the opposte propertes are true. A summary of the etals of the characterstcs propertes of the waves n the three rngs are gven Table III. (g) For all the three rngs F, G an E nstablty occurs only for the wave wth wave velocty λ = λ 3 <V 0, the ust rft velocty. For λ = λ 1,λ 2 >V 0 the waves are ampe an the ampng rate mnshes as λ V 0 ncreases. In fact the wave wth λ = λ 1 s almost unampe. In ths paper, we have consere sphercal ust grans wth unform sze. In future, we shall stuy the effect of ust sze strbuton on nonlnear ust acoustc wave, as n the astrophyscal plasmas, t has an mportant effect on ust acoustc wave (Havnes et al., 1990; Havnes et al., 1996). References Burns, J.A. an Schaffer: 1989, Nature 337, 340. Bharuthram, R. an Shukla, P.K.: 1992, Planet. Space Sc. 40, 973. Barkan, A., Merlno, R.L. an D Angelo, N.: 1995, Phys. Plasmas 2, 3563. Drane, B.T. an Salpeter, E.E.: 1979, Astrophys J. 231, 77. e Angels, U., Formsano an Gorano, M.: 1988, J. Plasma Phys. 40, 399. D Angelo, N.D.: 1990, Planet. Space Sc. 38, 1143. e Angels, U.: 1992, Phys. Scr. 45, 465. Goertz, C.K. an Shan, L.: 1988, Geophys. Res. Lett. 15, 84. Goertz, C.K.: 1989, Rev. Geophys. 27, 271. Ghosh, S., Sarkar, S., Khan, M. an Gupta, M.R.: 2000, Planet. Space Sc. 48, 609. Horyan, M.: 1996, Annu. Rev. Astron. Astrophys. 34, 383. Havnes, O., Aslaksen, T. an Melanso, F.: 1990, J. Geophys. Res. 95, 383.
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